We seek to develop a method that incorporates quantum mechanical dynamics into the analysis with the dependency on initial conditions rather then the dynamical equations that govern the evolution of the trajectories defined by the expectation values.
2.2. Representation Theory in the Coordinates
Following our previous works [
5,
6], we now define a new set of Hermitian operators
such that the commutation relations between the momenta and the operators
are given by a form as follows, in correspondence with Equation (
26), given in our previous work [
5],
with a tensor metric operator in a general operator-valued Hermitian form.
The canonical commutation relations, Equation (
27), applied to the left side of Equation (
38) results in
Therefore, one may define the following relation (for a full discussion of this point see [
5])
as a local relation between the two sets of coordinates, suggesting that, as a result of such a definition, one may think formally of a transformation between the two coordinate bases,
and
[
5,
6].
Applying the Heisenberg picture for the variables corresponding to
, and substituting Equation (
38), results in
such that it satisfies the following relation with the momentum operator
substituting Equation (
18), which is closely related to the form given in definition Equation (
22) [
5].
In analogy to our previous work [
6] and DeWitt’s work [
15] introducing DeWitt’s point transformation formula for the quantum transformation law for the momentum operators [
15], to designate general point transformations between the momentum operator
p expressed in the
space representation and the
expressed in the
space representation, covariant under point transformations, we define
Substituting Equation (
39)–(
40), a local relation between the two sets of coordinates
and
, the momentum operator
becomes
so that
has the same form (up to a factor of two) as the reduced connection form in the classical case (Equation (
10)) and is consistent with Equation (
36), correctly symmetrized so as to make it Hermitian.
Note that is time-dependent via the dependency of on which gives the origin for the underlying geometry involved with the space representation.
Furthermore, Equation (
17) is substituted so that Equation (
43) for the momentum operator
implies the following relation
The relations between the two momentum operators, substituting
in Equation (
43), are
Then, the commutation relations result in
The relations between the momenta and the velocities
are (from Equations (
45) and (
42)), so that
is consistent with Equations (
17) and (
18) and closely related with the definition in Equations (
22) and (
23) [
5], along with the definition for
(Equation (
43)).
Therefore, in analogy to our previous work [
6] where we worked with two space representations, motivated by the work of Horwitz et al. [
1] on the stability of classical Hamiltonian systems by geometrical methods, where a coordinate space,
, which is called the
Hamilton manifold, is endowed with a connection form
(Equation (
10)) and what is called the dynamical curvature. It is not uniquely defined in terms of the original manifold
, which is called the
Gutzwiller manifold.
To rigorously complete our mathematical formulation, we define two space representations, in analogy to our previous work [
6], corresponding to the
space representation and the
space representation:
Definition 2. (Gutzwiller representation)
Let be a Hilbert space corresponding to a given quantum mechanical system, and let be the self-adjoint geometric Hamiltonian generating the evolution of the system as before. Let the position and momentum operators, x and p, be as before and satisfy the canonical commutation relations (CCR).
The position space wave functions, expressed in the space representationwhere , are then said to be the Gutzwiller representation. Definition 3. (Hamilton representation)
Let be a Hilbert space corresponding to a given quantum mechanical system, and let be the self-adjoint geometric Hamiltonian generating the evolution of the system as before.
Let the position and momentum operators, x and p, be as before, and let the position and momentum operators, y and , be as beforeand satisfy the commutation relationsThe position space wavefunctions are said to be the Hamilton representation, when expressed in the space representationwhere denotes the volume element, and the integration is to be carried out over the entire range of coordinate values . Note that the Hamilton representation carries an integration over the underlying manifold which is induced by the tensor metric operator .
2.3. Local Existence of an Underlying Ricci Flow in the Quantum Theory
In this section, encouraged to find a method which brings the quantum mechanical dynamics into the analysis of quantum chaos, we then construct a generalized form of the dynamical equations in the Heisenberg picture governing the evolution of the tensor metric operator
, Equation (
37), in the form of a “geometric flow” [
16,
17,
18].
Our so-obtained construction carries a structure closely related in form to the classical Ricci flow equations, which are time-evolution equations, i.e., equations of motion equal to the first-order time derivatives of .
We start by following our previous analysis of the Heisenberg picture for the tensor metric operator
as a Heisenberg dynamical variable (Equation (
37)), satisfying the Heisenberg’s form for the equations of motion expressed in the
representation. Following our previous formal definition for a transformation between the two space representations
and
, Equation (
37) expressed in the
space representation results in
Note that the momentum is expressed in the
space representation, covariant under point transformations between the two space representations [
6].
Then, we express the momentum operator
by the substitution of Equation (
48) to relate the momentum
with the velocity field expressed in the
space representation. A relation between the two dynamical evolutions of the operators
and
is obtained,
Next, we define the “
Riemann curvature” operator in the formal form
expressed in the
space representation, to be a Hermitian function of the position operator
x (since
is a c-number-valued function of the
x coordinate base) acting on
, equivalent to the classical form of the Riemann curvature tensor in the
coordinate base (associated with the geodesic deviation equation in the Hamilton manifold, accounting for the dynamical curvature in Equation (
14) [
1]).
We define, as well, the “
Ricci curvature” operator in a formal form to be
classically equivalent in form to the Ricci curvature tensor obtained by taking the contraction of the second and fourth indices of the Riemann curvature tensor.
We suggest, as an analogy to the classical case, a formal construction for a quantum mechanical “Ricci flow” in a generalized form, associated with the general operator-valued Hermitian geometric Hamiltonian of the form (
29) following the Heisenberg algebra with the geometric Hamiltonian. The quantum mechanical “Ricci flow” equation is expressed by the representation theory in the
coordinates on the Hilbert space
corresponding to a given quantum mechanical system.
Substituting Equations (
55) and (
56) in Equation (
53) results in
where
is the Ricci operator defined in Equation (
56).
Next, given the structure of Equation (
57), we define a new operator in a generalized form to be an additional term to the Ricci operator in the quantum mechanical dynamical Equation (
57), formally defined as follows
and
is a general symmetric operator acting on
.
In what follows, motivated by the structure implied by Equation (
57), we focus on the topic of geometrization of the quantum mechanical dynamics in the study of geometric flows. We discuss only Ricci flow here but analogues of these results may apply more broadly.
First, we define the adiabatic flow, then, the tensor metric operator is used to define a “distance”, to be described below, within the operator space, so that it becomes a metric space.
The general form of
associated with the evolution generated by the geometric Hamiltonian in the form of Equation (
29), following Equation (
57), is
such that, expressed in the
space representation (substitution of Equation (
48))
is obtained. Next, we define an adiabatic quantum mechanical evolution. We start by following operator-valued analysis of the
operator Equation (
59) which results in
where we assume that
is invertible, so that
, while
and
denotes the n-by-n identity matrix.
Substituting Equation (
48), i.e.,
, in Equation (
61) satisfies the following relation
Define
to be a general operator acting on
, giving Equation (
58), such that
In the special case of Equation (
59), the
operator, yielding Equation (
62), results in
We now consider the case of a bounded
operator, considered to be small, in a manner which ensures that
Equation (
65) is in a structure that bounds the momentum
p in the case that the right-hand side of Equation (
65) is bounded (i.e., manifolds with bounded geometry, to be described below).
Substituting Equation (
65) in Equation (
60) results in
which ensures that
is small as well.
Further operator-valued analysis results in
implying that the tensor metric operator is
adiabatically changing in a sense that the commutation relation of the rate of change of the tensor metric operator with the position operator is small and
.
In our recent work on the subject of quantum mechanical Hamiltonian Evolution on a Finsler Manifold, generated by the self-adjoint geometric Hamiltonian
[
6], we found results with dynamical equations governing the evolution of the trajectories defined by the expectation values of the position expressed in the
space representation in a structure of two terms, where the first term is the quantum mechanical form of a “geodesic flow”, with a connection form as in Equation (
44) (up to a factor of two), and with a second term which is an essentially quantum effect. Note that this term, which was derived from the adiabatic case of a slowly changing momentum operator in a Finsler geometry, is in the form of
and carries a structure closely related to the first term in Equation (
67).
The evolving dynamical equation of
derived in [
6] results in
where two of the indices
are contracted with momentum, following a behavior such that the underlying geodesic flow, as exhibited by the expectation values, is subjected to the presence of an additional kind of “force” term, which is an essentially quantum effect, and has the consequence of contributing to the forces driving the system [
6,
19,
20,
21,
22].
This term accounts for the underlying “geometric flow” altering the Hamilton manifold of the underlying geometry [
6]. Classically, the sign of the eigenvalues of this term may contribute to the local stability properties of the geodesic flow on the Hamilton manifold, reflecting different behaviors of its geometric evolution [
6].
Therefore, our study here on the quantum mechanical Ricci flow is essentially by its nature a geometric flow on a Finsler geometry (a perturbed Ricci flow on a Finsler geometry). The term of Equation (
67), in this sense, may affect the stability properties of the perturbed Ricci flow [
23,
24].
We now turn to the following definitions.
Definition 4. (Adiabatic Flow)
Let be a Hilbert space corresponding to a given quantum mechanical system and let be the self-adjoint geometric Hamiltonian generating the evolution of the system as before.
Let the Riemann curvature operator , the Ricci curvature operator and the operator be as before.
Define to be a bounded operator acting on , considered to be small, as before.
Let the quantum mechanical system at a given time undergo irreversible evolution such that the Hermiticity of the tensor metric operator is no longer preserved.
Then, an adiabatic evolution is defined to beA generalized quantum mechanical adiabatic flow equation readsso that and are functions of the tensor metric operator acting on (to be described below in Equation (75)). Classically, following the canonical quantization (Dirac’s correspondence principle),
becomes (
is the Poisson bracket)
Then, having defined the adiabatic flow, we define “distance” within the operator space and establish the perturbed Ricci flow as the underlying geometry induced by the operator metric space.
Definition 5. (Quantum Mechanical Ricci Flow)
Let be a Hilbert space corresponding to a given quantum mechanical system and let be the self-adjoint geometric Hamiltonian generating the evolution of the system as before.
Let the Riemann curvature operator , the Ricci curvature operator and the operator be as before.
Let the adiabatic evolution and the adiabatic flow be as before.
Define the operator space χ to be a metric space with a distance function for each such that denotes curves from into χ byfor all states in n-dimensional , expressed in the Hamilton representation as before. Denote the space of continuous curves for which this norm is finite by for a time interval , where lies in the space . Define the trajectory Φ corresponding to an initial curve to be [19,20]i.e., is the set of reached in the course of the evolution of the system from an initial . The trajectory is then said to be the quantum mechanical geometric flow within the space. Given any metric on n-manifold , a generalized quantum mechanical Ricci flow equation within the metric operator space readsso that and are tensors acting on and has at least a perturbed unique smooth solution on , so that is a small perturbation of the Ricci flow solution (ϵ is positive or negative), and is a smooth one-parameter family of metrics on the n-manifold . We now turn to the final step, which suggests a definition for “local instability” in the quantum theory.
This notion relies on the work of Bahuaud et al. [
23,
24,
25], proving that the Ricci flow for complete metrics with bounded geometry depends continuously on initial conditions for a finite time with no loss of regularity [
24]. Their work leads to a general theorem (Theorem 3.6. in [
24]) about convergence stability around any given converging flow trajectory in any geometric flow, i.e., it applies not only to the Ricci flow but to more general geometric flows as well.
We suggest relating the stability of the quantum mechanical system with the principle of convergence stability for geometric flows in the operator metric space, which is the combination of the continuous dependence of the flow on initial conditions, with the stability of fixed points [
23,
24].
We first express
in the
coordinate base, substituting Equation (
48) in Equation (
67), expressed in the Hamilton representation
such that the generalized
quantum mechanical Ricci flow equation within the metric operator space
reads
We identify this geometric flow, under the adiabatic flow constraint, as the perturbed Ricci flow, where the perturbation term acts as an additional geometric correction.
We now invoke the the work of Bahuaud et al. [
23,
24,
25], studying the convergence stability for the Ricci flow on manifolds with bounded geometry and we apply their theorem (3.6.) [
24] below.
The Banach spaces on which we let these operators act are little
spaces. We work in the topology of little
spaces of metrics as in [
23,
24,
25].
Suppose that the geometric flow Equation (
76) is any ungauged geometric flow which admits a gauging, i.e., the addition of a term
tangent to the diffeomorphism orbit and such that
is an elliptic geometric operator which is admissible in the sense of [
25] (see discussions in [
24,
26,
27]).
Theorem 1. (Convergence Stability)
Let be a Hilbert space corresponding to a given quantum mechanical system and let be the self-adjoint geometric Hamiltonian generating the evolution of the system as before.
Let the quantum mechanical Ricci flow (which is said to be the perturbed Ricci flow) be as before and the geometric flow equation readsgiven that is any metric of a given geometry on n-manifold . Let the quantum mechanical Ricci flow be an ungauged geometric flow. If the addition of a term tangent to the diffeomorphism orbit and is an elliptic geometric operator which is admissible in the sense of [25], the quantum mechanical Ricci flow is said to be admitting a gauging. Let be any stationary point of the quantum mechanical Ricci flow equation, both ungauged and gauged flows [23,24,25], which is strictly stable. Let be any metric which is the initial point of a solution to the quantum mechanical Ricci flow (ungauged flow). Then for any metric g sufficiently near to in an appropriate weighted norm for some and , the ungauged flow converges to and the entire trajectory remains close to for all t. The quantum mechanical system is then said to be stable in along the evolution.
However, if at some time , the quantum mechanical system is then said to be locally unstable in at and the entire trajectory may not remain close to for all t.